We note that such a result is not direct and in fact surprising given the general lack of continuity in the space of information structures under weak convergence (which is indeed the notion considered in our positive result when we will, under weak convergence, approach a dirac delta measure by a sequence of Gaussian measures with decreasing variances): [1]} shows that the optimal cost in partially observable stochastic control is not weakly continuous on the space of measurement channels in general, and in fact this negative result also applies to setwise perturbations of measurement channels [1]} even when the costs considered are bounded continuous and all spaces considered are compact (see also [3]}). To obtain our positive result in this paper, we will consider a cautious construction where Lusin's theorem will be utilized twice, first to approximate functions and second to approximate stochastic kernels with their continuous and weakly continuous versions, respectively.
